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Theorem prnz 4496
Description: A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)
Hypothesis
Ref Expression
prnz.1 𝐴 ∈ V
Assertion
Ref Expression
prnz {𝐴, 𝐵} ≠ ∅

Proof of Theorem prnz
StepHypRef Expression
1 prnz.1 . . 3 𝐴 ∈ V
21prid1 4484 . 2 𝐴 ∈ {𝐴, 𝐵}
32ne0ii 4122 1 {𝐴, 𝐵} ≠ ∅
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  wne 2969  Vcvv 3383  c0 4113  {cpr 4368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-v 3385  df-dif 3770  df-un 3772  df-nul 4114  df-sn 4367  df-pr 4369
This theorem is referenced by:  opnz  5130  propssopi  5162  fiint  8477  wilthlem2  25144  upgrbi  26320  wlkvtxiedg  26866  shincli  28738  chincli  28836  spr0nelg  42513  sprvalpwn0  42520
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